2.14 The role of diffeomorphisms

As in other discrete formulations, the spatial diffeomorphism group cannot be realized exactly on the lattice, and the only obvious symmetries of a cubic lattice are discrete rotations and overall translations. The commutator computation described above indicates that one may be able recover the diffeomorphism invariance in a suitable continuum limit. Corichi and Zapata [83] have suggested the presence of a residual diffeomorphism symmetry in the lattice theory, under which, for example, all non-intersecting Wilson loop lattice states would be identified.

One can try to interpret the lattice theory as a manifestly diffeomorphism-invariant construction, with the lattice representing an entire diffeomorphism equivalence class of lattices embedded in the continuum [146]. In order to make this interpretation consistent, one should modify the functional form of either the Hamiltonian or the measure, in such a way that the commutator of two lattice Hamiltonians vanishes, as a → 0.

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